Performance Analysis of Reliability Growth Models using Supervised Learning Techniques

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1 Inernaional Journal of Scienific & Technology Research Volume 1,Issue 1,Feb 2012 ISSN Performance Analysis of Reliabiliy Growh Models using Supervised Learning Techniques Y Vamsidhar, P Samba Siva Raju, T Ravi Kumar Absrac Sofware reliabiliy is one of a number of aspecs of compuer sofware which can be aken ino consideraion when deermining he qualiy of he sofware. Building good reliabiliy models is one of he key problems in he field of sofware reliabiliy. A good sofware reliabiliy model should give good predicions of fuure failure behavior, compue useful quaniies and be widely applicable. Sofware Reliabiliy Growh Models (SRGMs) are very imporan for esimaing and predicing sofware reliabiliy. An ideal SRGM should provide consisenly accurae reliabiliy esimaion and predicion across differen projecs. However, ha here is no single such model which can obain accurae resuls for differen cases. The reason is ha he performance of SRGMs highly depends on he assumpions on he failure behavior and he applicaion daa-ses. In oher words, many models may be shown o perform well wih one failure daa-se, bu bad wih he oher daa-se. Thus, combining some individual SRGMs han single model is helpful o obain a more accurae esimaion and predicion. SRGM parameers are esimaed using he leas square esimaion (LSE) or Maximum Likelihood Esimaion (MLE). Several combinaional mehods of SRGMs have been proposed o improve he reliabiliy esimaion and predicion accuracy. The AdaBoosing algorihm is one of he mos popular machine learning algorihms. An AdaBoosing based Combinaional Model (ACM) is used o combine he several models. The key idea of his approach is ha we selec several SRGMs as he weak predicors and use an AdaBoosing algorihm o deermine he weighs of hese models for obaining he final linear combinaional model. In his paper, he Finess and Predicion of various Sofware Reliabiliy Growh Models (SRGMs) can be compared wih AdaBoosing based Combinaional Model (ACM) wih he help of Maximum likelihood esimaion o esimae he model parameers. Index Terms Sofware Reliabiliy, Sofware Reliabiliy Growh Models (SRGMs), AdaBoosing Algorihm, Leas Square Esimaion, Maximum Likelihood Esimaion. 1 INTRODUCTION S OFTWARE Reliabiliy Engineering is defined as quaniaive sudy of he operaional behavior of sofwarebased sysems wih respec o user requiremens concerning reliabiliy. The demand for sofware sysems has recenly increased very rapidly. The reliabiliy of sofware sysems has become a criical issue in he sofware sysems indusry. Wih he 90 s of he previous cenury, compuer sofware sysems have become he major source of repored failures in many sysems. Sofware is considered reliable if anyone can depend on i and use i in criical sysems. The imporance of sofware reliabiliy will increase in he years o come, specifically in he fields of aerospace indusry, saellies, and medicine applicaions. The process of sofware reliabiliy sars wih sofware esing and gahering of es resuls, afer ha, he phase of building a reliabiliy model. In general, he concep of reliabiliy can be defined as he probabiliy ha a sysem will perform is inended funcion during a period of running ime wihou any failure. IEEE defines sofware reliabiliy as he probabiliy of failurefree sofware operaions for a specified period of ime in a Y Vamsidhar is working as Assoc. Professor & HOD, CSE Dep., Swarnandhra College of Engg. & Technology, Narsapur, W.G.Dis. (A.P), India, PH scecsehod@gmail.com P Samba Siva Raju is working as Ass. Professor, CSE Dep., VISIT Engg. College, Tadepalligudem, W.G.Dis. (A.P), India, PH p.raj1987@gmail.com T Ravi Kumar is working as Ass. Professor, CSE Dep., VISIT Engg. College, Tadepalligudem, W.G.Dis. (A.P), India, PH engineer.ravikumar@gmail.com specified environmen. In oher words, sofware reliabiliy can be viewed as he analysis of is failures, heir causes and effecs. Sofware reliabiliy is a key characerisic of produc qualiy. Mos ofen, specific crieria and performance measures are placed ino reliabiliy analysis, and if he performance is below a cerain level, failure occurred. Mahemaically, he reliabiliy funcion R() is he probabiliy ha a sysem will be successfully operaing wihou failure in he inerval from ime 0 o ime [1], R() = P(T>), where >=0 T is a random variable represening he failure ime or ime-o-failure (i.e. he expeced value of he lifeime before a failure occurs). R() is he probabiliy ha he sysem's lifeime is larger han (), he probabiliy ha he sysem will survive beyond ime, or he probabiliy ha he sysem will fail afer ime. From above Equaion, we can conclude ha failure probabiliy F(), unreliabiliy funcion of T is: F() = 1 R() = P(T<=) If he ime-o-failure random variable T has a densiy funcion f(), hen he reliabiliy can be measured as following R() = f(x) dx Where f(x) represens he densiy funcion for he random variable T. Consequenly, he hree funcions R(), F() and f() are closely relaed o one anoher. IJSTR

2 2 SUPERVISED LEARNING TECHNIQUES 2.1 Boosrap The Boosrap procedure is a general purpose sample-based saisical mehod which consiss of drawing randomly wih replacemen from a se of daa poins. I has he purpose of assessing he saisical accuracy of some esimae, say S(Z), over a raining se Z = {z 1, z 2,..., z N}, wih z i = (x i, y i). To check he accuracy, he measure is applied over he B sampled versions of he raining se. The Boosrap algorihm[10] is as follows: Boosrap Algorihm Inpu: Training se Z={z 1,z 2,,z N}, wih z i=(x i,y i). B, number of sampled versions of he raining se. Oupu: S(Z), saisical esimae and is accuracy. Sep 1: for n=1 o B a) Draw, wih replacemen, L < N samples from he raining se Z, obaining he nh sample Z *n. b) For each sample Z*n, esimae a saisic S(Z *n ). Sep 2: Produce he boosrap esimae S(Z), using S(Z *n ) wih n = {1,...,B}. Sep 3: Compue he accuracy of he esimae, using he variance or some oher crierion. We sar wih he raining se Z, obaining several versions Z *n (boosrap samples). For each sampled version, we compue he desired saisical measure S(Z *n ). 2.2 Bagging The Bagging echnique[10] consiss of Boosrap aggregaion. Le us consider a raining se Z ={z 1, z 2,..., z N}, wih z i = (x i, y i) for which we inend o fi a regression model, obaining a predicion f(x) a inpu x. Bagging averages his predicion over a collecion of boosrap samples, hereby reducing is variance. For classificaion purposes, he Bagging algorihm is as follows. Bagging Algorihm for Classificaion Inpu: Z = {z 1, z 2,..., z N}, wih z i = (x i, y i) as raining se. B, number of sampled versions of he raining se. Oupu: H(x), a classifier suied for he raining se. Sep 2: Produce he final classifier as a voe of Hn wih n = {1,...,B} H(x)= sign (Σ B n=1 Hn(x)) As compared o he process of learning a classifier in a convenional way, ha is, sricly from he raining se, he Bagging approach increases classifier sabiliy and reduces variance. 2.3 Boosing The Boosing[10] procedure is similar o Boosrap and Bagging. The firs was proposed in 1989 by Schapire and is as follows. Boosing Algorihm for Classificaion Inpu: Z = {z 1, z 2,..., z N}, wih z i = (x i, y i) as raining se. Oupu: H(x), a classifier suied for he raining se. Sep 1: Randomly selec, wihou replacemen, L1 < N samples from Z o obain Z *1 ; rain weak learner H1 on Z *1. Sep 2: Selec L2 < N samples from Z wih half of he samples misclassified by H1 o obain Z *2 ; rain weak learner H2 on i. Sep 3: Selec all samples from Z ha H1 and H2 disagree on; rain weak learner H3, using hree samples. Sep 4: Produce final classifier as a voe of he hree weak learners H(x) = sign (Σ 3 n=1 H n(x)) 2.4 AdaBoosing A very promising well known used boosing algorihm is AdaBoos [2]. The idea behind adapive boosing is o weigh he daa insead of (randomly) sampling i and discarding i. In recen years, ensemble echniques, namely boosing algorihms, have been a focus of research. The AdaBoos algorihm is a well-known mehod o build ensembles of classifiers wih very good performance. I has been shown empirically ha AdaBoos wih decision rees has excellen performance, being considered he bes off-he-shelf classificaion algorihm. AdaBoosing is a commonly used ML algorihm which can combine several weak predicors ino a single srong predicor for highly improving he esimaion and predicion accuracy. I has been applied wih grea success o several benchmark Machine Learning (ML) problems using raher simple learning algorihms, in paricular decision rees. AdaBoosing Algorihm AdaBoosing is a commonly used machine learning algorihm for consrucing a srong classifier f(x) as linear combinaion of weak classifiers h (x). Sep 1: for n=1 o B 1 a) Draw, wih replacemen, L < N samples from he raining AdaBoosing calls a weak classifier h (x) repeaedly in a se Z, obaining he nh sample Z *n. series of rounds =1, 2,..,T. For each call a disribuion of b) For each sample Z *n, learn classifier Hn. weighs α is updaed in he daa-se for he classificaion. The IJSTR T f ( x) h ( x)

3 algorihm akes as inpu a raining se (x 1, y 1),(x 2, y 2),.,(x n, y n) where x iєx, y iє{-1,+1} (x 1, y 1) where each x i belongs o some domain or insance space X, and each label y i is in some label se Y. One of he main ideas of he algorihm is o mainain a disribuion or se of weighs over he raining se. Given (x 1, y 1),,(x n, y n) where x i єx, y i є{-1,+1} Iniialize weighs D (i) = 1/n Ierae =1,,T: Train weak learner using disribuion D Ge weak classifier: h : X R Choose R Updae o D +1(i) from D (i) Where Z is a normalizaion facor (chosen so ha D +1 will be a disribuion), and Oupu Final Classifier 3 EXISTING SYSTEM D ( i)exp( yih ( xi )) D 1( i) Z 1 1 ln 0 2 H ( x) sign( 3.1 Sofware Reliabiliy Growh Models A sofware reliabiliy growh model (SRGM) describes he mahemaical relaionship of finding and removing fauls o improve sofware reliabiliy. A SRGM performs curve fiing of observed failure daa by a pre-specified model formula, where he parameers of he model are found by saisical echniques like maximum likelihood mehod. The model hen esimaes reliabiliy or predics fuure reliabiliy by differen forms of exrapolaion. Afer he firs sofware reliabiliy growh model was proposed by Jelinski and Moranda in 1972, here have been numerous reliabiliy growh models following i. These models come under differen classes, e.g. exponenial failure ime class of models, Weibull and Gamma failure ime class of models, infinie failure caegory models and Bayesian models. These models are based on prior assumpions abou he naure of failures and he probabiliy of individual failures occurring. There is no reliabiliy growh model ha can be generalized for all possible sofware projecs, alhough here is evidence of T 1 h ( x)) models ha are beer suied o cerain ypes of sofware projecs. An imporan class of SRGM ha has been widely sudied is he NonHomogeneous Poisson Process (NHPP). I forms one of he main classes of he exising SRGM, due o i is mahemaical racabiliy and wide applicabiliy. NHPP models are useful in describing failure processes, providing rends such as reliabiliy growh and he faul - conen. SRGM consider he debugging process as a couning process characerized by he mean value funcion of a NHPP. Sofware reliabiliy can be esimaed once he mean value funcion is deermined. Model parameers are usually deermined using eiher Maximum Likelihood Esimae (MLE) or leas-square esimaion mehods. NHPP based SRGM are generally classified ino wo groups[13]. The firs group conains models, which use he execuion ime (i.e., CPU ime) or calendar ime. Such models are called coninuous ime models. The second group conains models, which use he number of es cases as a uni of he faul deecion period. Such models are called discree ime models, since he uni of he sofware faul deecion period is counable. A es case can be a single compuer es run execued in an hour, day, week or even monh. Therefore, i includes he compuer es run and lengh of ime spen o visually inspec he sofware source code. An imporan class of SRGM ha has been widely sudied is he NonHomogeneous Poisson Process (NHPP). I forms one of he main classes of he exising SRGM; due o i is mahemaical racabiliy and wide applicabiliy. NHPP models are useful in describing failure processes, providing rends such as reliabiliy growh and he faul - conen. SRGM consider he Debugging process as a couning process characerized by he mean value funcion of a NHPP. Sofware reliabiliy can be esimaed once he mean value funcion is deermined. Model parameers are usually deermined using eiher Maximum Likelihood Esimae (MLE) or leas-square esimaion mehods. Sofware reliabiliy growh models wih a model are formulaed by using a Non-Homogeneous Poisson Process (NHPP). Using he model, he mehod of daa analysis for he sofware reliabiliy measuremen will be developed. SRGM parameers are esimaed by using he leas square esimaion (LSE) or maximum likelihood esimaion (MLE) mehod and using acual sofware failure daa, numerical resuls are obained. In he exising sysem, he sofware reliabiliy growh model parameers are esimaed using leas square esimaion o obain he numerical resuls. 3.2 Limiaions of Exising Sysem I is generally considered o have less desirable opimaliy properies han maximum likelihood. I can be quie sensiive o he choice of saring values. IJSTR

4 I can generae accurae resuls for daa of sufficienly large samples. 4 PROPOSED WORK & METHODOLOGY In he proposed sysem, he reliabiliy growh model parameers are esimaed using maximum likelihood esimaion o ge accurae resuls, o overcome he problems of LSE, we use MLE. Maximum likelihood provides a consisen approach o parameer esimaion problems. This means ha maximum likelihood esimaes can be developed for a large variey of esimaion siuaions. For example, hey can be applied in reliabiliy analysis o censored daa under various censoring models. MLE has many opimal properies in esimaion, i.e. Sufficiency: complee informaion abou he parameer of ineres conained in is MLE esimaor. Consisency: rue parameer value ha generaed he daa recovered asympoically, i.e. for daa of sufficienly large samples. Efficiency: lowes-possible variance of he parameer esimaes achieved asympoically. 4.1 Seleced Models for Parameer Esimaion and Comparison Crierion A sofware reliabiliy growh model characerizes how he reliabiliy of ha sofware varies wih execuion ime. The radiional sofware reliabiliy models are se of echniques ha apply probabiliy heory and saisical analysis o sofware reliabiliy. A reliabiliy model specifies he general form of he dependence of he failure process on he principal facors ha affecs i. The following five models are seleced as he candidae and comparison models, which have been widely used by many researchers in he field of sofware reliabiliy modeling[6]. (1) Goel-Okumoo model (GO Model) : M1 This model is proposed by he Goel and Okumoo, one of he mos popular non-homogeneous poisson process (NHPP) model in he field of sofware reliabiliy modeling. I assumes failures occur randomly and ha all fauls conribue an equally o oal unreliabiliy. When a failure occurs, i assumes ha he fix is perfec, hus he failure rae improves coninuously in ime. a(1-exp(-r)), a>0,r>0 (2) Musa-Okumoo model (MO Model) : M2 I is also a nonhomogeneous poisson process wih an inensiy funcion ha decreases exponenially as failures occur. The exponenial rae of decrease reflecs he view ha he earlier discovered failures have a greaer impac on reducing he failure inensiy funcion han hose encounered laer. 1/a*ln(1+ar) (3) Delayed S-Shaped Model : M3 Yamada presened a delayed S-shaped SRGM incorporaing he ime delay beween faul deecion and faul correcion. The Delayed S-Shaped model is a modificaion of he NHPP o obain an S-shaped curve for he cumulaive number of failures deeced such ha he failure rae iniially increased and laer decays. a(1-(1+r)exp(-r)) (4) Infleced S-Shaped Model : M4 Ohba proposed an infleced S-shaped model o describe he sofware failure-occurrence phenomenon wih muual dependency of deecing fauls. a(1-exp(-r)) (1+c*exp(-r)) (5) Generalized GO Model : M5 In GO model, he failure occurrence rae per faul is ime independen, however since he expeced number of remaining fauls decreases wih ime, he overall sofware failure inensiy decreases wih ime. In mos real-life esing scenarios, he sofware failure inensiy increases iniially and hen decreases. The generalized GO model was proposed o capure his increasing/decreasing naure of he failure inensiy. a(1-exp(-r c )) In hese NHPP models, usually parameer a usually represens he mean number of sofware failures ha will evenually be deeced, and parameer r represens he probabiliy ha a failure is deeced in a consan period [13]. In his, wo real failure daa-ses are seleced, Ohba and Wood [5], which are popular and frequenly used for comparison of SRGM. 4.2 AdaBoosing based Combinaional Model (ACM) Many SRGMs may resul in esimaion or predicion bias since heir underlying assumpions are no consisen wih he characerisics of he applicaion daa. To reduce his bias, combining several differen SRGMs ogeher in linear or nonlinear manner is a common and applicable mehod. Many approaches are proposed o deermine he weigh assignmen for he combinaional models, such as equal weigh, neuralnework, geneic programming and ec. In his, an absrac descripion of how o use AdaBoosing o obain he dynamic weighed linear combinaional model (ACM) of several SRGMs is shown as follows[3] Inpu: 1. Le M()=u(x 1,x 2, x k x S, ) denoes differen SRGMs (such as GO model). M() is he cumulaed fauls deeced a ime, S is he number of he parameers of M(), x k is he k-h parameer of M(), k=1,2,3,...s; IJSTR

5 2. The failure daa-se D0 is denoed by ( 1, m 1), ( 2, m 2), ( j, m j)... ( n, m n). Iniialize: Where n is he daa number of D 0, m j is he cumulaed fauls deeced a j, j=1...n. Sep1: Selecing M differen SRGMs (denoed by M m(), m=1...m) as he candidae models for he ACM. Sep2: The original weigh se of D 0 can be denoed by K 0 = {k k 0n}, where k 0j is iniialized by 1/n. Circulaion: Sep3: New raining daa se D i (i=1...p, P is he raining rounds) are generaed by repeaedly random sampling from D 0 according o is corresponding weigh se K i={k i1 k ij k in}. If here are some same daa in D i, only one of hem will be reserved in our approach. Hence he daa number of Di may no be n. Sep4: D i is used o esimae he parameers of each M m() in he i-h raining round. Then he finess funcion (Noaion 1) of M m() can be deermined by D 0. The candidae model whose value of finess funcion is he smalles is chosen as he seleced model (denoed by M is()) in his round, i=1...p. Sep5: If M1 is() denoes he esimaion form of M is(), he loss funcion L is (Noaion 2) of M is () can be calculaed by he fiing resuls of M1 is () wih D 0. Then K i can be updaed as K i+1 by L is (Noaion 3). The basic weigh β is of M is () also can be deermined by L is(noaion 4). Sep6: Performing Sep3-Sep5 repeaedly unil i=p, and hen urning ino Sep7. Oupu: Sep7: Finally, a combinaional linear model is obained as follows: M ACM() = Σ P i=1 W is M is() The combinaion weigh W is of he seleced model M is () is f (β is) (Noaion 5), ha is, W is is a funcion of he basic weigh β is. Noaion 1: Finess funcion (FF) can be defined according o he esimaion mehod of he parameers of hese candidae SRGMs. The general mehods o esimae he parameers of SRGMs are leas-squares esimaion (LSE) and maximum likelihood esimaion (MLE). If Maximum likelihood Esimaion is used, FF equaion is FF =1/ log (ML) Where ML is he maximum likelihood funcion derived in he nex secion. Noaion 2: L is = Σ n j=1 k ij * L j is L j is = AE j is / D en Where AE j is = m j - M1 is ( j), Den =max {AE j is} Noaion 3: K i+1 = { K i+1, j }, K i+1, j = k ij * L j is / (Σ n j=1 k ij * L j is ) Noaion 4: β is = L j is /(1- L j is) Noaion 5: W is = f(β is ) = log (1/ β is ) / Σ p i=1 log 1/ β is 4.3 Parameer Esimaion Once he analyic expression for he mean value funcion is derived, i is required o esimae he parameers in he mean value funcion, which is usually carried ou by using Maximum likelihood Esimaion echnique[11]. Maximum Likelihood Esimaion: Once a model is specified wih is parameers, and daa have been colleced, one is in a posiion o evaluae is goodness of fi, ha is, how well i fis he observed daa. Goodness of fi is assessed by finding parameer values of a model ha bes fis he daa, a procedure called parameer esimaion. The general mehods o esimae he parameers are leassquares esimaion (LSE) and maximum likelihood esimaion (MLE). Fiing a proposed model o acual faul daa involves esimaing he model parameers from he real es daa ses. Here we employ he mehod of MLE o esimae he parameers a and r. All parameers of differen Reliabiliy models can be esimaed by he mehod of MLE. For example, suppose ha a and r are deermined for he observed daa pairs: ( 0, m 0), ( 1, m 1), ( 2, m 2),.,( n, m n). Then he likelihood funcion for he parameers a and r in he models wih m() is given by ML = ( ) ( )! exp[ (m( ) m( ))] Where, mj is he cumulaive number of fauls deeced by j es cases (j=1,2,3,.,n), j is accumulaed number of es run execued o deec m j fauls, m( j) is he expeced mean number of fauls deeced by he nh es case. ge Taking he naural logarihm of he above equaion, we lnml = Σ n j=1(m j-m j-1) ln[m( j)-m( j-1)] - {(m( j)-m( j-1)} IJSTR

6 Σn j=1 ln[(m j-m j-1)!] The values of hese parameers ha maximize he sample likelihood are known as he Maximum Likelihood Esimaor. 4.4 Analysis of Daa Daa Se #1: The firs daa se employed was from he paper by Ohba[5] for a PL/I daabase applicaion sofware sysem consising of approximaely LOC. Over he course of 19 weeks, CPU hours were consumed, and 328 sofware fauls were removed. Alhough his is an old daa-se, we feel i is insrucive o use i because i allows direc comparison wih he work of ohers who have used i. Daa Se #2: The second daa se presened by Wood[5] from a subse of producs for four separae sofware releases a Tandem Compuers Company. Wood repored ha he specific producs & releases are no idenified, and he es daa ses have been suiably ransformed in order o avoid confidenialiy issues. Here we only use Release 1 for illusraions. Over he course of 20 weeks, CPU hours were consumed, and 100 sofware fauls were removed. 5 Experimenal Resuls In order o implemen he fiing and predicion performance of he five seleced models and ACM by using maximum likelihood esimaion o esimae he parameers of models can be shown as follows. 5.1 Finess and Predicion Performance of five Models and ACM wih failure daases The Figures (Fig 5.1 & Fig 5.2) represens he Finess and Predicion graphs of various SRGMs and ACM respecively. Here he failure daa-se Ohba [5] is aken for he comparison purpose. Fig 5.2 Predicion comparison beween Models and ACM wih Ohba Daase 6 Conclusion and Fuure Scope In his paper, he Finess and Predicion of various Sofware Reliabiliy Growh Models (SRGMs) can be compared wih AdaBoosing based Combinaional Model (ACM) wih he help of Maximum likelihood esimaion o esimae he model parameers. From he resuls, he fiing and predicion performance of ACM is beer compare wih individual reliabiliy growh models wih real failure daa-ses. The furher enhancemens hose are possible for his, Examining he saisical significance of he esimaion and predicion resuls of he ACM and Comparing wih he oher combinaion approaches such as Geneic-based Combinaional Model (GCM)[9], Dynamic Weighed Combinaional Model (DWCM)[7] ec. References [1] Hoang Pham, Sysem Sofware Reliabiliy. Springer Series in Reliabiliy Engineering.. [2] Jiri Maas and Jan S ochman, AdaBoos, Cenre for Machine Percepion, Czech Technical Universiy, Prague. [3] Haifeng Li, Min Zeng, and Minyan Lu, Exploring AdaBoosing Algorihm for Combining Sofware Reliabiliy Models,ISSRE [4] X. Cai, M. R. Lyu. Sofware Reliabiliy Modeling wih Tes Coverage Experimenaion and Measuremen wih a Faul-Toleran Sofware Projec. ISSRE, 2007: [5] C. Y. Huang, S. Y. Kuo and M. R. Lyu. An assessmen of esing-effor dependen sofware reliabiliy growh models. IEEE Transacions on Reliabiliy, 2007, 56(2): [6] Lyu, M. R, Nikora, A. Applying Reliabiliy Models More Effecive. IEEE Sofware, 1992, 9(4): [7] Y. S. Su, C. Y. Huang. Neural-nework based approaches for sofware reliabiliy esimaion using dynamic weighed combinaional models. The Journal of Sysems and Sofware, 2007, 80: Fig 5.1 Finess comparison beween Models and ACM wih Ohba Daase [8] C. J. Hsu, C. Y. Huang. Reliabiliy analysis using weighed combinaional models for web-based sofware. WWW 2009, IJSTR

7 [9] Eduardo Oliveira Cosa, Silvia R. Vergilio, Aurora Pozo, Gusavo Souza. Modeling sofware reliabiliy growh wih Geneic Programming. ISSRE, 2005: 1-10 [10]Arur Ferreira, Survey on Boosing algorihms for supervised and semi-supervised learning. Insiue of Telecommunicaions. [11] Aasia Quyoum, Mehraj Ud - Din Dar, Improving Sofware Reliabiliy using Sofware Engineering Approach, Inernaional Journal of Compuer Applicaions ( ) Volume 10 No.5, November [12] S. Yamada, J. Hishiani, and S. Osaki, Sofware reliabiliy growh model wih Weibull esing effor: a model and applicaion, IEEE Trans. Reliabiliy, vol. R-42, pp , [13]Alan Wood, Sofware reliabiliy growh models. Tandem Compuers. IJSTR